Newton’s first interpolation formula
Let in equidistant points , where h – step of interpolation , values are given for function . It is required to pick up polynomial of degree not above n , satisfying to conditions (1).
Let's enter finite differences for sequence of values :
Conditions (1) are equivalent to the equalities:
Lowering the transformations resulted in [1 ], we’ll finally receive Newton’s first interpolation formula :
where – number of steps of interpolation from beginning point up to point х .
The formula (3) is expedient for using for interpolation of function in locality of beginning point , where q modulo a little.
In special cases we have:
at n = 1 – formula of linear interpolation :
at n = 2 – formula of square-law or parabolic interpolation :
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